IJMMS 31:1 (2002) 23–29
PII. S0161171202110398
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ASYMPTOTIC HÖLDER ABSOLUTE VALUES
E. MUÑOZ GARCIA
Received 12 October 2001
We prove that asymptotic Hölder absolute values are Hölder equivalent to classical absolute values. As a corollary we obtain a generalization of Ostrowski’s theorem and a classical theorem by E. Artin. The theorem presented implies a new, more flexible, definition of
classical absolute value.
2000 Mathematics Subject Classification: 12J20, 12J10, 16W80, 13J99.
1. Introduction. Asymptotic Hölder absolute values generalize the notions of classical absolute value and of Hölder absolute value. A Hölder absolute value (HAV) satisfies an approximate triangle inequality and multiplicative property. More precisely,
let C1 ≥ 1 and C2 ≥ 1. A (C1 , C2 )-Hölder absolute value on a ring R is a mapping
· : R → R+ satisfying:
(HAV1) for x ∈ R, x = 0 x = 0;
(HAV2) for x, y ∈ R, x + y ≤ C2 (x + y);
(HAV3) for x, y ∈ R, C1−1 xy ≤ xy ≤ C1 xy.
It is known that HAV on a ring are Hölder equivalent to a classical ones. More
precisely, we have the following theorem (see [2]).
Theorem 1.1 (Hölder rigidity). Let · : R → R+ be a (C1 , C2 )-Hölder absolute value
on a commutative ring R with unit element. There exists an absolute value on R, | · | :
R → R+ , which is (C1α , α)-Hölder equivalent to · with α = log2 (2C2 ), that is, for x ∈ R,
C1−α |x|α ≤ x ≤ C1α |x|α .
(1.1)
Moreover, | · | can be defined by
1/na
.
|x| = lim x n n→+∞
(1.2)
For a ring R with unity, a real constant C2 ≥ 1, and a function C1 (·, ·) defined on
]1, +∞[×N taking values in [1, +∞[, we define a (C1 , C2 )-asymptotic Hölder absolute
value (AHAV) on R,
| · | : R → R+ ,
satisfying the three following axioms:
(AHAV1) |x| = 0 if and only if x = 0;
(AHAV2) for x, y ∈ R, |x + y| ≤ C2 (|x| + |y|);
(1.3)
24
E. MUÑOZ GARCIA
(AHAV3) for γ > 1 and n ≥ 2 there is a constant C1 (γ, n) > 1 such that for x1 , . . . ,
xn ∈ R,
γ −1 γ
γ
γ −1
· · · xn ≤ x1 · · · xn ≤ C1 (γ, n)x1 · · · xn ,
C1 (γ, n)−1 x1 (1.4)
and L = limn→∞ (1/n) log C1 (γ, n) < +∞.
We prove the following theorem.
Theorem 1.2. Let R be a commutative ring with unity. Let C2 ≥ 1 be a real constant,
α = 1/ log2 (2C2 ), and · a (C1 , C2 )-AHAV on R. We have the following dichotomy:
(i) if
1
lim log C1 (γ, n) = 0,
(1.5)
n→∞ n
then · α is a classical absolute value on R;
(ii) if
1
0 < L = lim log C1 (γ, n) < +∞,
n→∞ n
(1.6)
then · α is a Hölder absolute value on R, more precisely, it is (eLα , α)-Hölder
equivalent to an absolute value on R.
As a result of Theorem 1.2(i), we can define classical absolute values as AHAV with
C2 = 1 having a sequence of constants (C1 (γ, n))n growing sub-exponentially, that is,
lim
1
n→∞ n
log C1 (γ, n) = 0.
(1.7)
This is far more flexible than the classical definition.
Note that, in general, Hölder equivalence is a metric property which is stronger than
the usual topological equivalence, for example, {0} ∪ {1/n; n ≥ 1} and {0} ∪ {1/2n ;
n ≥ 1} are homeomorphic, but not Hölder equivalent.
Corollary 1.3. Consider | · | : R → R+ satisfying
(AV1) |x| = 0 if and only if x = 0,
(AV2) for x, y ∈ R, |x + y| ≤ |x| + |y| then,
(AV3) for x, y ∈ R, |xy| = |x||y| is equivalent to:
(AV3 ) for γ > 1 and n ≥ 2 there is a constant C1 (γ, n) > 1 such that for x1 , . . . , xn ∈
R,
γ −1 γ
γ
γ −1
· · · xn ≤ x1 · · · xn ≤ C1 (γ, n)x1 · · · xn C1 (γ, n)−1 x1 (1.8)
with limn→∞ (1/n) log C1 (γ, n) = 0.
Our theorem gives a generalization for discrete rings of Artin’s theorem [1].
Corollary 1.4. If · is a (1, C2 )-AHAV over a discrete field F , there exists an
absolute value | · | and an exponent α, such that for all x in F , xα = |x|.
Also, our theorem implies a generalization of Ostrowski’s theorem [3] for classical
absolute values (C1 = C2 = γ = 1) over Z.
ASYMPTOTIC HÖLDER ABSOLUTE VALUES
25
Corollary 1.5. If · is a (C1 , C2 )-AHAV over Z normalized, so that 1 = 1, then
· is (eLα , α)-Hölder equivalent to a p-adic absolute value | · |p or to | · |∞ or to the
trivial absolute value, with α = 1/ log2 (2C2 ).
Remarks. (1) The constant C1 (γ, n) in the definition of AHAV can be chosen to
satisfy the inequality
n
(1.9)
C1 (γ, n) ≤ C1 γ 1/([log2 n]+1) , 2 ,
where [a] denotes the integer part of a.
(2) Let C2 ≥ 1 and let | · | : R → R+ be a (C1 , C2 )-AHAV on R. If limγ→1 C1 (γ, 2) = C1 <
+∞, then | · | is a (C1 , C2 )-Hölder absolute value.
(3) If R is a ring on which a (C1 , C2 )-AHAV | · | is defined, then R is a discrete ring
for the topology defined by | · |.
1.1. Weak subadditive lemma. We prove a generalization of a classical lemma on
subadditive sequences (which might be of independent interest).
Definition 1.6. The real sequence (bm )m∈N is weakly subadditive if
(i) for γ > 1 and k ≥ 1, there is a constant K(γ, k) > 0 such that for m1 , . . . , mk ∈ N,
bm1 +···+mk ≤ γ
k
i=1
bmi + K(γ, k);
(1.10)
(ii) for γ > 1, we have K ∗ (γ) = limk→∞ (1/k)K(γ, k) < +∞.
Lemma 1.7. If (bm )m∈N is weakly subadditive, then
lim
m→∞
bm
bm
= lim
.
m→∞ m
m
(1.11)
Proof. Fix n ≥ 1. For any m ∈ Z, we consider the Euclidean division
m = nq + r ,
Now,
0 ≤ r < n.
(1.12)
bm = bnq+r ≤ γ qbn + br + K(γ, q + 1).
(1.13)
bnq+r
bm
br
q
q + 1 K(γ, q + 1)
=
≤γ
bn +
+
.
m
nq + r
nq + r
nq + r
nq + r
q+1
(1.14)
Dividing by m,
Taking the upper limit when m → ∞,
bm
bn
1
≤γ
+ 0 + K ∗ (γ).
m→∞ m
n
n
lim
(1.15)
That is, for all q ≥ 1,
lim
m→∞
bn 1 ∗
bm
≤γ
+ K (γ).
m
n
n
(1.16)
26
E. MUÑOZ GARCIA
Now, taking the lower limit on the right side when n → ∞,
lim
m→∞
bm
bn
≤ γ lim
.
m
n→∞ n
(1.17)
This holds for all γ > 1, thus making γ → 1,
lim
m→∞
bm
bm
≤ lim
,
m
m→∞ m
lim
m→∞
bm
bm
= lim
.
m
m→∞ m
(1.18)
1.2. Proof of Theorem 1.1
Lemma 1.8. Define | · | : R → R+ by |x| = xα . Then, | · | is a (C1α , 2)-AHAV
on R.
Proof. (AHAV1) |x| = 0 if and only if x = 0 if and only if x = 0.
(AHAV2) |x + y| = x + yα ≤ (2C2 )α (max(x, y))α ≤ 2(xα + yα ) =
2(|x| + |y|).
(AHAV3) For all γ > 1 and for all n ≥ 2 there is a constant C1 (γ, n)α > 1 such that
for all x1 , . . . , xn in R,
γ −1
−α γ −1
x1 · · · xn C1 (γ, n)
γ
γ
≤ x1 · · · xn ≤ C1 (γ, n)α x1 · · · xn .
(1.19)
Lemma 1.9. Let x ∈ R and define the real sequence (an )n∈N by an = |x n |. The
1/n
sequence (an ) is converging and
e−L |x| ≤ lim a1/n
≤ eL |x|,
n
n→∞
(1.20)
where L = limn→∞ (1/n) log C1 (γ, n) < +∞.
Proof. Let bm = log am . The sequence {bm } is weakly subadditive, since for all
γ > 1 and for all k ≥ 1 there is a constant K(γ, k) = (C1 (γ, k))α , such that
bm1 +···+mk ≤ γ
k
i=1
bmi + log K(γ, k),
(1.21)
and for all γ > 1,
lim
k→+∞
1
log K(γ, k) < +∞.
k
(1.22)
bm
bm
= lim
.
m→∞ m
m
(1.23)
Therefore, by Lemma 1.7,
lim
m→∞
1/n
1/n
Thus, to prove the convergence of {an }, we only have to prove that {an } is
bounded.
Let γ > 1, for n ∈ N there is C1 (γ, n)α satisfying
C1 (γ, n)−α |x|n/γ ≤ x n ≤ C1 (γ, n)α |x|nγ .
(1.24)
ASYMPTOTIC HÖLDER ABSOLUTE VALUES
27
Taking nth roots,
≤ C1 (γ, n)α/n |x|γ .
C1 (γ, n)−α/n |x|1/γ ≤ a1/n
n
(1.25)
Since L = limn→∞ (1/n) log C1 (γ, n) < +∞, we obtain
≤ eαL |x|γ .
e−αL |x|1/γ ≤ lim a1/n
n
n→∞
(1.26)
This inequality holds for any γ > 1. Taking the limit when γ → 1,
≤ eαL |x|.
e−αL |x| ≤ a1/n
n
(1.27)
Now we define that |·| : R → R+ by |0| = 0 and that |x| = limn→∞ |x n |1/n for x ≠ 0.
Lemma 1.10. The function | · | : R → R+ defined as above is an absolute value on R.
Moreover, if limn→+∞ (1/n) log C1 (γ, n) = 0, then |x| = xα for all x ∈ R.
Proof. From Lemma 1.9, if limn→∞ (1/n) log C1 (γ, n) = 0, we obtain
|x| ≤ |x| ≤ |x|.
(1.28)
That is, |x| = xα .
It is clear that, |x| = 0 if and only if x = 0. Next we check the multiplicative property.
For γ > 1 and for n ≥ 2 there exists C1 (γ, 2)α > 1, such that for n ∈ N and x, y in R,
γ −1 γ −1
C1 (γ, 2)−α x n y n γ γ
≤ x n y n ≤ C1 (γ, n)α x n y n .
(1.29)
Taking nth roots and passing to the limit when n → +∞, we obtain
|x|γ
−1
|y|γ
−1
≤ |xy| ≤ |x|γ |y|γ .
(1.30)
Taking the limit when γ → 1, we have the desired multiplicative property.
Finally, we have to check the triangle inequality. This is a corollary of the following
general proposition that gives an equivalent, apparently weaker, definition of absolute
value.
Proposition 1.11. Let R be a commutative ring with unity. Let | · | : R → R+ be a
function satisfying the following three properties:
(A1) |x| = 0 if and only if x = 0;
(A2) (approximate triangle inequality) there exists a real constant B > 0, such that for
all x, y in R, |x + y| ≤ B(|x| + |y|);
(A3) for x, y in R, |xy| = |x||y|.
Then, | · | is an absolute value on R, that is, | · | satisfies the triangle inequality.
Lemma 1.12. For x, y ∈ R,
|x + y| ≤ B |x| + |y| ≤ 2B max |x|, |y| .
(1.31)
28
E. MUÑOZ GARCIA
Lemma 1.13. Let | · | : R → R+ , such that for x, y ∈ R,
|x + y| ≤ M max |x| , |y| ,
(1.32)
for some positive constant M. Then for x1 , x2 , . . . , xn ∈ R,
n
xi ≤ M [log2 n]+1 max xi ,
1≤i≤n
i=1 (1.33)
where [a] denotes the integer part of a.
Proof. Let m = [log2 n] + 1 and complete the sequence (xi )1≤i≤n into (xi )1≤i≤2m
adjoining 0 elements.

m 
m−1
m
2
2 2


xi ≤ M max 
xi xi ,

i=1
i=1 i=2m−1 +1 
m−1

m−2
m−2
m
2
3·2
2
2
 (1.34)

xi xi xi xi ≤ M 2 max 
,
,
,

i=2m−2 +1 i=2m−1 +1 i=3·2m−2 +1 i=1
≤ · · · ≤ M m max xi .
1≤i≤2m
Lemma 1.14. Let Z be the image of Z in R. For n ∈ N,
|n̄| ≤ 2n|1|.
(1.35)
Proof. We use Lemma 1.13 with M = 2 and | · | = | · |. Take m = [log2 n] + 1,
n ≤ 2m ≤ 2n, and xi = 1 for 1 ≤ i ≤ n. We have
n
xi ≤ 2m |1| ≤ 2n|1|.
|n| = (1.36)
i=1
Lemma 1.15. Let Z be the image of Z in R. For n ∈ N,
|n̄| ≤ n.
(1.37)
k
nk = n̄ ≤ 2nk |1|,
(1.38)
Proof. Using Lemma 1.14,
and |n̄k |1/k ≤ 21/k n|1|1/k . Taking k → +∞, we have |n̄| ≤ n.
Proof of Proposition 1.11. Let x, y ∈ R and n ≥ 1. Let m = [log2 n] + 1. Using
Lemmas 1.12 and 1.14, we have
n
n
i n−i (x + y)n = y
x
i=0 i
(1.39)
n
m
i n−i .
≤ (B) max x y
0≤i≤n i
ASYMPTOTIC HÖLDER ABSOLUTE VALUES
29
Now using Lemma 1.14,
n (x + y)n ≤ (2B)m max |x|i |y|n−i
0≤i≤n i n
|x|i |y|n−i
≤ (2B)m max
0≤i≤n i
n
n
m
≤ (2B)
|x|i |y|n−i
i
i=0
n
≤ (2B)m |x| + |y| .
Finally,
1/n
≤ (2B)(1/n)([log2 n]+1) |x| + |y| ,
|x + y| = (x + y)n (1.40)
(1.41)
and passing to the limit n → +∞ we get the sharp triangle inequality |x +y| ≤ |x|+|y|.
Proof of Theorem 1.2.
Case 1. Assume limn→∞ (1/n) log C1 (γ, n) = 0. By Lemma 1.8, for all x, y in R we
have
|x + y| = |x + y| ≤ 2 |x| + |y| ≤ 4 max |x|, |y| = 4 max |x|, |y| .
(1.42)
Therefore, by Proposition 1.11, the function | · | satisfies the triangle inequality.
Case 2. Assume 0 < L = limn→∞ (1/n) log C1 (γ, n) < +∞. From Lemma 1.9, for any
x in R,
(1.43)
e−αL |x| ≤ |x| ≤ eαL |x|.
Therefore,
|x + y| ≤ eαL |x + y| ≤ 2eαL |x| + |y| ≤ 2e2αL |x| + |y| .
(1.44)
Thus by Proposition 1.11, the function | · | satisfies the triangle inequality, it is an
absolute value, and · α is (eLα , α)-equivalent to | · |.
References
[1]
[2]
[3]
E. Artin, Theory of Algebraic Numbers, George Striker, Göttingen, 1959.
E. Muñoz Garcia, Hölder absolute values are equivalent to classical ones, Proc. Amer. Math.
Soc. 127 (1999), no. 7, 1967–1971.
A. Ostrowski, Über einige Lösungen der Funktionalgleichung φ(x)·φ(y) = φ(x ·y), Acta
Math. 41 (1914), 271–284 (German).
E . Muñoz Garcia: Department of Mathematics, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, CA 90095-1555, USA
E-mail address: munoz@math.ucla.edu